Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Returns data

The long version of Velliv medium risk data runs from January 2007 to April 2024 (incl).
For January 2007 to May 2012 no low risk and high risk funds existed. For this period the medium risk data is copied into the other funds.

The short version runs from June 2012 to April 2024.

Velliv returns are including bonus and “DinKapital.
PFA returns are including”KundeKapital”.

Summary of log-returns

The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)

vmr vhr vmrl pmr phr mmr mhr vmr_phr vhr_pmr
Min. : 0.901 0.877 0.901 0.924 0.886 0.912 0.882 0.893 0.899
1st Qu.: 0.996 0.994 0.995 0.999 0.994 0.997 0.995 0.996 0.996
Median : 1.010 1.012 1.007 1.008 1.012 1.009 1.013 1.011 1.011
Mean : 1.006 1.007 1.005 1.005 1.008 1.006 1.008 1.007 1.006
3rd Qu.: 1.021 1.027 1.020 1.015 1.025 1.018 1.025 1.022 1.021
Max. : 1.070 1.088 1.070 1.043 1.079 1.054 1.082 1.073 1.065

Ranking

Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.924 pmr 0.999 pmr 1.013 mhr 1.008 phr 1.027 vhr 1.088 vhr
0.912 mmr 0.997 mmr 1.012 phr 1.008 mhr 1.025 phr 1.082 mhr
0.901 vmr 0.996 vmr 1.012 vhr 1.007 vhr 1.025 mhr 1.079 phr
0.901 vmrl 0.996 vhr_pmr 1.011 vmr_phr 1.007 vmr_phr 1.022 vmr_phr 1.073 vmr_phr
0.899 vhr_pmr 0.996 vmr_phr 1.011 vhr_pmr 1.006 vhr_pmr 1.021 vmr 1.070 vmr
0.893 vmr_phr 0.995 mhr 1.010 vmr 1.006 vmr 1.021 vhr_pmr 1.070 vmrl
0.886 phr 0.995 vmrl 1.009 mmr 1.006 mmr 1.020 vmrl 1.065 vhr_pmr
0.882 mhr 0.994 phr 1.008 pmr 1.005 pmr 1.018 mmr 1.054 mmr
0.877 vhr 0.994 vhr 1.007 vmrl 1.005 vmrl 1.015 pmr 1.043 pmr

Correlations and covariance

Correlations

vmr vhr pmr phr
vmr 1.000 0.997 0.961 0.964
vhr 0.997 1.000 0.951 0.967
pmr 0.961 0.951 1.000 0.977
phr 0.964 0.967 0.977 1.000

Covariances

vmr vhr pmr phr
vmr 0.001 0.001 0 0.001
vhr 0.001 0.001 0 0.001
pmr 0.000 0.000 0 0.000
phr 0.001 0.001 0 0.001

Compare pension plans

Risk of loss

Risk of loss at least as big as row name in percent for a single period (year).

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 32.333 33.000 30.167 31.667 31.167 32.167 31.833 31.667
5 2.500 4.167 0.667 3.167 1.500 3.500 2.667 2.500
10 0.167 0.500 0.000 0.167 0.000 0.333 0.167 0.167
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 28.833 29.500 22.667 29.333 25.667 29.333 29 27.333
5 1.833 3.167 0.833 2.500 1.167 2.667 2 1.833
10 0.000 0.333 0.000 0.000 0.000 0.167 0 0.000
25 0.000 0.000 0.000 0.000 0.000 0.000 0 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0 0.000

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 40.333 40.667 37.833 38.500 39.167 39.500 39.167 39.667
5 0.500 2.333 0.000 1.333 0.000 1.667 0.833 0.500
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
90 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Worst ranking for loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
33.000 vhr 4.167 vhr 0.500 vhr 0 vmr 0 vmr 0 vmr 0 vmr
32.333 vmr 3.500 mhr 0.333 mhr 0 vhr 0 vhr 0 vhr 0 vhr
32.167 mhr 3.167 phr 0.167 vmr 0 pmr 0 pmr 0 pmr 0 pmr
31.833 vmr_phr 2.667 vmr_phr 0.167 phr 0 phr 0 phr 0 phr 0 phr
31.667 phr 2.500 vmr 0.167 vmr_phr 0 mmr 0 mmr 0 mmr 0 mmr
31.667 vhr_pmr 2.500 vhr_pmr 0.167 vhr_pmr 0 mhr 0 mhr 0 mhr 0 mhr
31.167 mmr 1.500 mmr 0.000 pmr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
30.167 pmr 0.667 pmr 0.000 mmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
29.500 vhr 3.167 vhr 0.333 vhr 0 vmr 0 vmr 0 vmr 0 vmr
29.333 phr 2.667 mhr 0.167 mhr 0 vhr 0 vhr 0 vhr 0 vhr
29.333 mhr 2.500 phr 0.000 vmr 0 pmr 0 pmr 0 pmr 0 pmr
29.000 vmr_phr 2.000 vmr_phr 0.000 pmr 0 phr 0 phr 0 phr 0 phr
28.833 vmr 1.833 vmr 0.000 phr 0 mmr 0 mmr 0 mmr 0 mmr
27.333 vhr_pmr 1.833 vhr_pmr 0.000 mmr 0 mhr 0 mhr 0 mhr 0 mhr
25.667 mmr 1.167 mmr 0.000 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
22.667 pmr 0.833 pmr 0.000 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
40.667 vhr 2.333 vhr 0 vmr 0 vmr 0 vmr 0 vmr 0 vmr
40.333 vmr 1.667 mhr 0 vhr 0 vhr 0 vhr 0 vhr 0 vhr
39.667 vhr_pmr 1.333 phr 0 pmr 0 pmr 0 pmr 0 pmr 0 pmr
39.500 mhr 0.833 vmr_phr 0 phr 0 phr 0 phr 0 phr 0 phr
39.167 mmr 0.500 vmr 0 mmr 0 mmr 0 mmr 0 mmr 0 mmr
39.167 vmr_phr 0.500 vhr_pmr 0 mhr 0 mhr 0 mhr 0 mhr 0 mhr
38.500 phr 0.000 pmr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
37.833 pmr 0.000 mmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Chance of min gains

Chance of gains of at least x percent for a single period (year).
x values are row names.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 67.667 67.000 69.833 68.333 68.833 67.833 68.167 68.333
5 1.167 3.833 0.167 3.667 0.500 3.333 2.167 1.333
10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 71.167 70.500 77.333 70.667 74.333 70.667 71.000 72.667
5 7.833 13.500 3.667 11.500 5.500 12.500 9.667 8.167
10 0.667 1.833 0.167 1.167 0.333 1.333 0.833 0.833
25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 59.667 59.333 62.167 61.500 60.833 60.5 60.833 60.333
5 3.333 8.333 0.167 7.167 1.333 7.5 5.167 3.500
10 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
25 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
50 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000
100 0.000 0.000 0.000 0.000 0.000 0.0 0.000 0.000

Best ranking for gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
69.833 pmr 3.833 vhr 0 vmr 0 vmr 0 vmr 0 vmr
68.833 mmr 3.667 phr 0 vhr 0 vhr 0 vhr 0 vhr
68.333 phr 3.333 mhr 0 pmr 0 pmr 0 pmr 0 pmr
68.333 vhr_pmr 2.167 vmr_phr 0 phr 0 phr 0 phr 0 phr
68.167 vmr_phr 1.333 vhr_pmr 0 mmr 0 mmr 0 mmr 0 mmr
67.833 mhr 1.167 vmr 0 mhr 0 mhr 0 mhr 0 mhr
67.667 vmr 0.500 mmr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
67.000 vhr 0.167 pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
77.333 pmr 13.500 vhr 1.833 vhr 0 vmr 0 vmr 0 vmr
74.333 mmr 12.500 mhr 1.333 mhr 0 vhr 0 vhr 0 vhr
72.667 vhr_pmr 11.500 phr 1.167 phr 0 pmr 0 pmr 0 pmr
71.167 vmr 9.667 vmr_phr 0.833 vmr_phr 0 phr 0 phr 0 phr
71.000 vmr_phr 8.167 vhr_pmr 0.833 vhr_pmr 0 mmr 0 mmr 0 mmr
70.667 phr 7.833 vmr 0.667 vmr 0 mhr 0 mhr 0 mhr
70.667 mhr 5.500 mmr 0.333 mmr 0 vmr_phr 0 vmr_phr 0 vmr_phr
70.500 vhr 3.667 pmr 0.167 pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
62.167 pmr 8.333 vhr 0 vmr 0 vmr 0 vmr 0 vmr
61.500 phr 7.500 mhr 0 vhr 0 vhr 0 vhr 0 vhr
60.833 mmr 7.167 phr 0 pmr 0 pmr 0 pmr 0 pmr
60.833 vmr_phr 5.167 vmr_phr 0 phr 0 phr 0 phr 0 phr
60.500 mhr 3.500 vhr_pmr 0 mmr 0 mmr 0 mmr 0 mmr
60.333 vhr_pmr 3.333 vmr 0 mhr 0 mhr 0 mhr 0 mhr
59.667 vmr 1.333 mmr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
59.333 vhr 0.167 pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

MC risk percentiles

Risk of loss at least as big as row name in percent from first to last period.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 17.71 18.08 10.00 11.89 7.65 7.23 7.11 8.27
5 9.25 11.04 3.51 6.44 2.25 2.58 2.60 2.78
10 4.54 6.57 1.19 3.42 0.62 0.90 0.75 0.85
25 0.47 0.88 0.09 0.35 0.02 0.02 0.04 0.03
50 0.02 0.03 0.01 0.00 0.00 0.00 0.00 0.00
90 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 4.21 4.55 2.33 3.44 0.64 0.58 0.60 0.62
5 2.12 2.64 1.13 1.67 0.30 0.18 0.18 0.23
10 1.08 1.32 0.57 0.73 0.11 0.11 0.08 0.06
25 0.10 0.21 0.15 0.06 0.02 0.01 0.00 0.01
50 0.00 0.00 0.04 0.00 0.00 0.00 0.00 0.00
90 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00
99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 14.05 15.07 9.03 9.83 4.83 4.39 4.46 5.75
5 6.01 8.08 2.22 4.34 0.64 1.03 1.01 1.15
10 1.97 3.53 0.29 1.66 0.02 0.23 0.14 0.12
25 0.00 0.07 0.00 0.02 0.00 0.00 0.00 0.00
50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
90 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Worst ranking for MC loss percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
18.08 vhr 11.04 vhr 6.57 vhr 0.88 vhr 0.03 vhr 0 vmr 0 vmr
17.71 vmr 9.25 vmr 4.54 vmr 0.47 vmr 0.02 vmr 0 vhr 0 vhr
11.89 phr 6.44 phr 3.42 phr 0.35 phr 0.01 pmr 0 pmr 0 pmr
10.00 pmr 3.51 pmr 1.19 pmr 0.09 pmr 0.00 phr 0 phr 0 phr
8.27 vhr_pmr 2.78 vhr_pmr 0.90 mhr 0.04 vmr_phr 0.00 mmr 0 mmr 0 mmr
7.65 mmr 2.60 vmr_phr 0.85 vhr_pmr 0.03 vhr_pmr 0.00 mhr 0 mhr 0 mhr
7.23 mhr 2.58 mhr 0.75 vmr_phr 0.02 mmr 0.00 vmr_phr 0 vmr_phr 0 vmr_phr
7.11 vmr_phr 2.25 mmr 0.62 mmr 0.02 mhr 0.00 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
4.55 vhr 2.64 vhr 1.32 vhr 0.21 vhr 0.04 pmr 0.01 pmr 0 vmr
4.21 vmr 2.12 vmr 1.08 vmr 0.15 pmr 0.00 vmr 0.00 vmr 0 vhr
3.44 phr 1.67 phr 0.73 phr 0.10 vmr 0.00 vhr 0.00 vhr 0 pmr
2.33 pmr 1.13 pmr 0.57 pmr 0.06 phr 0.00 phr 0.00 phr 0 phr
0.64 mmr 0.30 mmr 0.11 mmr 0.02 mmr 0.00 mmr 0.00 mmr 0 mmr
0.62 vhr_pmr 0.23 vhr_pmr 0.11 mhr 0.01 mhr 0.00 mhr 0.00 mhr 0 mhr
0.60 vmr_phr 0.18 mhr 0.08 vmr_phr 0.01 vhr_pmr 0.00 vmr_phr 0.00 vmr_phr 0 vmr_phr
0.58 mhr 0.18 vmr_phr 0.06 vhr_pmr 0.00 vmr_phr 0.00 vhr_pmr 0.00 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 90 ranking 99 ranking
15.07 vhr 8.08 vhr 3.53 vhr 0.07 vhr 0 vmr 0 vmr 0 vmr
14.05 vmr 6.01 vmr 1.97 vmr 0.02 phr 0 vhr 0 vhr 0 vhr
9.83 phr 4.34 phr 1.66 phr 0.00 vmr 0 pmr 0 pmr 0 pmr
9.03 pmr 2.22 pmr 0.29 pmr 0.00 pmr 0 phr 0 phr 0 phr
5.75 vhr_pmr 1.15 vhr_pmr 0.23 mhr 0.00 mmr 0 mmr 0 mmr 0 mmr
4.83 mmr 1.03 mhr 0.14 vmr_phr 0.00 mhr 0 mhr 0 mhr 0 mhr
4.46 vmr_phr 1.01 vmr_phr 0.12 vhr_pmr 0.00 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
4.39 mhr 0.64 mmr 0.02 mmr 0.00 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

MC gains percentiles

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 82.29 81.92 90.00 88.11 92.35 92.77 92.89 91.73
5 70.65 72.69 75.97 79.76 78.86 84.34 82.99 80.04
10 55.47 61.27 55.27 68.75 55.72 70.44 66.97 61.66
25 13.20 24.67 4.82 29.65 3.28 20.10 13.41 9.13
50 0.14 1.52 0.02 1.64 0.00 0.21 0.02 0.05
100 0.00 0.04 0.01 0.01 0.00 0.00 0.00 0.00
200 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.00
300 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00
400 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00
500 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00
1000 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 95.79 95.45 97.67 96.56 99.36 99.42 99.40 99.38
5 92.09 92.35 94.81 93.66 98.00 98.15 98.21 98.12
10 86.07 87.80 89.06 88.99 94.76 95.63 94.87 95.25
25 54.92 65.59 49.09 64.12 54.65 72.05 64.24 65.03
50 10.26 24.55 4.50 18.98 2.96 14.88 7.98 8.18
100 0.28 1.31 0.17 0.50 0.05 0.12 0.04 0.11
200 0.03 0.04 0.02 0.00 0.01 0.00 0.00 0.01
300 0.02 0.02 0.02 0.00 0.00 0.00 0.00 0.00
400 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00
500 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00
1000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0 85.95 84.93 90.97 90.17 95.17 95.61 95.54 94.25
5 72.53 74.85 75.33 81.58 82.78 87.77 85.46 82.92
10 56.07 62.19 52.17 69.19 58.26 74.22 69.32 62.77
25 13.98 25.16 4.79 29.60 4.08 21.63 13.74 9.72
50 0.40 2.19 0.00 2.08 0.00 0.21 0.05 0.01
100 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
200 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
300 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
400 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
500 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Best ranking for MC gains percentiles

Skewed \(t\)-distribution (sstd):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
92.89 vmr_phr 84.34 mhr 70.44 mhr 29.65 phr 1.64 phr 0.04 vhr
92.77 mhr 82.99 vmr_phr 68.75 phr 24.67 vhr 1.52 vhr 0.01 pmr
92.35 mmr 80.04 vhr_pmr 66.97 vmr_phr 20.10 mhr 0.21 mhr 0.01 phr
91.73 vhr_pmr 79.76 phr 61.66 vhr_pmr 13.41 vmr_phr 0.14 vmr 0.00 vmr
90.00 pmr 78.86 mmr 61.27 vhr 13.20 vmr 0.05 vhr_pmr 0.00 mmr
88.11 phr 75.97 pmr 55.72 mmr 9.13 vhr_pmr 0.02 pmr 0.00 mhr
82.29 vmr 72.69 vhr 55.47 vmr 4.82 pmr 0.02 vmr_phr 0.00 vmr_phr
81.92 vhr 70.65 vmr 55.27 pmr 3.28 mmr 0.00 mmr 0.00 vhr_pmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
0.01 vhr 0.01 vhr 0.01 vhr 0.01 vhr 0.01 vhr
0.01 phr 0.00 vmr 0.00 vmr 0.00 vmr 0.00 vmr
0.00 vmr 0.00 pmr 0.00 pmr 0.00 pmr 0.00 pmr
0.00 pmr 0.00 phr 0.00 phr 0.00 phr 0.00 phr
0.00 mmr 0.00 mmr 0.00 mmr 0.00 mmr 0.00 mmr
0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr 0.00 mhr
0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr
0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr

Standardized \(t\)-distribution (std):

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
99.42 mhr 98.21 vmr_phr 95.63 mhr 72.05 mhr 24.55 vhr 1.31 vhr
99.40 vmr_phr 98.15 mhr 95.25 vhr_pmr 65.59 vhr 18.98 phr 0.50 phr
99.38 vhr_pmr 98.12 vhr_pmr 94.87 vmr_phr 65.03 vhr_pmr 14.88 mhr 0.28 vmr
99.36 mmr 98.00 mmr 94.76 mmr 64.24 vmr_phr 10.26 vmr 0.17 pmr
97.67 pmr 94.81 pmr 89.06 pmr 64.12 phr 8.18 vhr_pmr 0.12 mhr
96.56 phr 93.66 phr 88.99 phr 54.92 vmr 7.98 vmr_phr 0.11 vhr_pmr
95.79 vmr 92.35 vhr 87.80 vhr 54.65 mmr 4.50 pmr 0.05 mmr
95.45 vhr 92.09 vmr 86.07 vmr 49.09 pmr 2.96 mmr 0.04 vmr_phr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
0.04 vhr 0.02 vmr 0.01 pmr 0.01 pmr 0 vmr
0.03 vmr 0.02 vhr 0.00 vmr 0.00 vmr 0 vhr
0.02 pmr 0.02 pmr 0.00 vhr 0.00 vhr 0 pmr
0.01 mmr 0.00 phr 0.00 phr 0.00 phr 0 phr
0.01 vhr_pmr 0.00 mmr 0.00 mmr 0.00 mmr 0 mmr
0.00 phr 0.00 mhr 0.00 mhr 0.00 mhr 0 mhr
0.00 mhr 0.00 vmr_phr 0.00 vmr_phr 0.00 vmr_phr 0 vmr_phr
0.00 vmr_phr 0.00 vhr_pmr 0.00 vhr_pmr 0.00 vhr_pmr 0 vhr_pmr

Normal distribution:

0 ranking 5 ranking 10 ranking 25 ranking 50 ranking 100 ranking
95.61 mhr 87.77 mhr 74.22 mhr 29.60 phr 2.19 vhr 0 vmr
95.54 vmr_phr 85.46 vmr_phr 69.32 vmr_phr 25.16 vhr 2.08 phr 0 vhr
95.17 mmr 82.92 vhr_pmr 69.19 phr 21.63 mhr 0.40 vmr 0 pmr
94.25 vhr_pmr 82.78 mmr 62.77 vhr_pmr 13.98 vmr 0.21 mhr 0 phr
90.97 pmr 81.58 phr 62.19 vhr 13.74 vmr_phr 0.05 vmr_phr 0 mmr
90.17 phr 75.33 pmr 58.26 mmr 9.72 vhr_pmr 0.01 vhr_pmr 0 mhr
85.95 vmr 74.85 vhr 56.07 vmr 4.79 pmr 0.00 pmr 0 vmr_phr
84.93 vhr 72.53 vmr 52.17 pmr 4.08 mmr 0.00 mmr 0 vhr_pmr
200 ranking 300 ranking 400 ranking 500 ranking 1000 ranking
0 vmr 0 vmr 0 vmr 0 vmr 0 vmr
0 vhr 0 vhr 0 vhr 0 vhr 0 vhr
0 pmr 0 pmr 0 pmr 0 pmr 0 pmr
0 phr 0 phr 0 phr 0 phr 0 phr
0 mmr 0 mmr 0 mmr 0 mmr 0 mmr
0 mhr 0 mhr 0 mhr 0 mhr 0 mhr
0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr 0 vmr_phr
0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr 0 vhr_pmr

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.005 0.007 0.005 0.008 0.005 0.007 0.007 0.006
s 0.027 0.034 0.019 0.030 0.023 0.031 0.028 0.027
nu 3.384 3.488 3.474 3.959 3.344 3.702 3.726 3.369
xi 0.699 0.708 0.770 0.737 0.716 0.714 0.715 0.709
R^2 0.993 0.992 0.994 0.996 0.993 0.993 0.994 0.993

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.013 0.015 0.012 0.014 0.012 0.015 0.014 0.013
s 0.032 0.040 0.027 0.035 0.029 0.037 0.033 0.033
nu 3.446 3.510 2.629 4.002 3.035 3.780 3.760 3.260
R^2 0.978 0.978 0.962 0.981 0.971 0.977 0.978 0.974

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
m 0.006 0.007 0.005 0.008 0.006 0.008 0.007 0.006
s 0.024 0.031 0.017 0.028 0.021 0.029 0.026 0.024
R^2 0.968 0.969 0.962 0.973 0.965 0.969 0.969 0.966

AIC and BIC

AIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -671.412 -603.343 -768.368 -622.474 -718.914 -617.207 -647.901 -672.555
std -628.069 -561.055 -728.782 -585.156 -674.537 -574.620 -605.443 -628.239
normal -646.514 -579.369 -743.179 -603.088 -692.459 -593.830 -624.670 -646.870

BIC

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
sstd -659.588 -591.520 -756.545 -610.651 -707.091 -605.384 -636.077 -660.732
std -616.246 -549.232 -716.959 -573.333 -662.714 -562.796 -593.620 -616.416
normal -634.691 -567.546 -731.356 -591.265 -680.636 -582.006 -612.847 -635.046

Kappa

Let \(\{X_{g,i}\}\) be Gaussian distributed with mean \(\mu\) and scale \(\sigma\).

Let \(\{X_{\nu,i}\}\) be \(t\)-distributed, scaled such that \(\mathbb{M}^{\nu}(1) = \mathbb{M}^{g}(1) = \sqrt{\frac{2}{\pi}} \sigma\).

Given \(n_g\), we want to determine and \(n_{\nu}^{*}\) such that

\[\text{Var}\left[\sum_i^{n_g} X_{g,i}\right] = \text{Var}\left[\sum_i^{n_{\nu}^{*}} X_{\nu,i}\right]\]

For iid. r.v \(\{X_i\}\):

\[S_n = X_1 + X_2 + \dots + X_n\] \[\mathbb{M}(n) = \mathbb{E}(\lvert S_n - \mathbb{E}(S_n)\rvert)\] Taleb’s convergence metric (\(\kappa\)):

The “rate” of convergence for \(n\) summands vs \(n_0\), i.e. the improved convergence achieved by \(n - n_0\) additional terms, is given by \(\kappa(n_0, n)\):

\[\kappa(n_0, n) = 2 - \dfrac{\log(n) - \log(n_0)}{\log\left(\frac{\mathbb{M}(n)}{\mathbb{M}(n_0)}\right)}\]

\(\kappa\)

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
0.16 0.16 0.16 0.13 0.15 0.14 0.13 0.17

\(n_{min}\)

What is the minimum value of \(n_{\nu}\), the number of observations from a given skewed \(t\)-distribution, we need to achieve the same degree of convergence as with \(n_g=30\) observations from a Gaussian distribution with the same mean and standard deviation?

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
58 55 55 48 58 51 50 57

Fit statistics ranking

Skewed \(t\)-distribution (sstd):

m ranking s ranking R^2 ranking
0.008 phr 0.019 pmr 0.996 phr
0.007 mhr 0.023 mmr 0.994 vmr_phr
0.007 vmr_phr 0.027 vhr_pmr 0.994 pmr
0.007 vhr 0.027 vmr 0.993 mmr
0.006 vhr_pmr 0.028 vmr_phr 0.993 mhr
0.005 vmr 0.030 phr 0.993 vmr
0.005 pmr 0.031 mhr 0.993 vhr_pmr
0.005 mmr 0.034 vhr 0.992 vhr

Standardized \(t\)-distribution (std):

m ranking s ranking R^2 ranking
0.015 vhr 0.027 pmr 0.981 phr
0.015 mhr 0.029 mmr 0.978 vmr
0.014 phr 0.032 vmr 0.978 vhr
0.014 vmr_phr 0.033 vhr_pmr 0.978 vmr_phr
0.013 vhr_pmr 0.033 vmr_phr 0.977 mhr
0.013 vmr 0.035 phr 0.974 vhr_pmr
0.012 mmr 0.037 mhr 0.971 mmr
0.012 pmr 0.040 vhr 0.962 pmr

Normal distribution:

m ranking s ranking R^2 ranking
0.008 phr 0.017 pmr 0.973 phr
0.008 mhr 0.021 mmr 0.969 vmr_phr
0.007 vhr 0.024 vhr_pmr 0.969 vhr
0.007 vmr_phr 0.024 vmr 0.969 mhr
0.006 vhr_pmr 0.026 vmr_phr 0.968 vmr
0.006 vmr 0.028 phr 0.966 vhr_pmr
0.006 mmr 0.029 mhr 0.965 mmr
0.005 pmr 0.031 vhr 0.962 pmr

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and-out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000

Skewed \(t\)-distribution (sstd):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 390.56 546.54 368.99 744.90 379.50 649.19 569.75 458.13
mc_s 162.67 310.60 106.60 358.90 98.41 230.77 195.66 154.95
mc_min 9.95 47.50 41.88 67.17 120.88 157.64 134.70 146.99
mc_max 1962.47 9418.03 964.46 3790.71 969.20 2350.77 2262.60 1819.14
dao_pct 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dai_pct 0.32 0.32 0.03 0.03 0.00 0.00 0.00 0.00

Standardized \(t\)-distribution (std):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 2281.79 4769.87 1795.80 3664.55 2345.79 4253.11 2979.65 3285.27
mc_s 1214.65 3240.61 1239.38 2099.78 31176.36 3565.88 1263.15 1921.35
mc_min 221.36 374.70 41.43 193.68 557.15 783.35 689.69 499.82
mc_max 16109.79 45965.08 94908.01 28483.53 3118787.05 295755.16 30086.97 59511.22
dao_pct 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dai_pct 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00

Normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
mc_m 441.92 627.78 365.42 789.64 403.75 711.18 614.07 498.41
mc_s 169.65 319.17 98.48 362.10 98.81 244.56 197.97 164.59
mc_min 90.51 80.75 123.03 120.90 165.90 188.76 136.86 160.53
mc_max 1791.46 3370.41 1109.65 3404.35 1102.37 2538.96 1961.78 1606.02
dao_pct 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
dai_pct 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00

Ranking

Skewed \(t\)-distribution (sstd):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
744.90 phr 98.41 mmr 157.64 mhr 9418.03 vhr 0 vmr 0.00 mmr
649.19 mhr 106.60 pmr 146.99 vhr_pmr 3790.71 phr 0 vhr 0.00 mhr
569.75 vmr_phr 154.95 vhr_pmr 134.70 vmr_phr 2350.77 mhr 0 pmr 0.00 vmr_phr
546.54 vhr 162.67 vmr 120.88 mmr 2262.60 vmr_phr 0 phr 0.00 vhr_pmr
458.13 vhr_pmr 195.66 vmr_phr 67.17 phr 1962.47 vmr 0 mmr 0.03 pmr
390.56 vmr 230.77 mhr 47.50 vhr 1819.14 vhr_pmr 0 mhr 0.03 phr
379.50 mmr 310.60 vhr 41.88 pmr 969.20 mmr 0 vmr_phr 0.32 vmr
368.99 pmr 358.90 phr 9.95 vmr 964.46 pmr 0 vhr_pmr 0.32 vhr

Standardized \(t\)-distribution (std):

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
4769.87 vhr 1214.65 vmr 783.35 mhr 3118787.05 mmr 0 vmr 0.00 vmr
4253.11 mhr 1239.38 pmr 689.69 vmr_phr 295755.16 mhr 0 vhr 0.00 vhr
3664.55 phr 1263.15 vmr_phr 557.15 mmr 94908.01 pmr 0 pmr 0.00 phr
3285.27 vhr_pmr 1921.35 vhr_pmr 499.82 vhr_pmr 59511.22 vhr_pmr 0 phr 0.00 mmr
2979.65 vmr_phr 2099.78 phr 374.70 vhr 45965.08 vhr 0 mmr 0.00 mhr
2345.79 mmr 3240.61 vhr 221.36 vmr 30086.97 vmr_phr 0 mhr 0.00 vmr_phr
2281.79 vmr 3565.88 mhr 193.68 phr 28483.53 phr 0 vmr_phr 0.00 vhr_pmr
1795.80 pmr 31176.36 mmr 41.43 pmr 16109.79 vmr 0 vhr_pmr 0.01 pmr

Normal distribution:

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_pct ranking dai_pct ranking
789.64 phr 98.48 pmr 188.76 mhr 3404.35 phr 0 vmr 0.00 pmr
711.18 mhr 98.81 mmr 165.90 mmr 3370.41 vhr 0 vhr 0.00 phr
627.78 vhr 164.59 vhr_pmr 160.53 vhr_pmr 2538.96 mhr 0 pmr 0.00 mmr
614.07 vmr_phr 169.65 vmr 136.86 vmr_phr 1961.78 vmr_phr 0 phr 0.00 mhr
498.41 vhr_pmr 197.97 vmr_phr 123.03 pmr 1791.46 vmr 0 mmr 0.00 vmr_phr
441.92 vmr 244.56 mhr 120.90 phr 1606.02 vhr_pmr 0 mhr 0.00 vhr_pmr
403.75 mmr 319.17 vhr 90.51 vmr 1109.65 pmr 0 vmr_phr 0.01 vmr
365.42 pmr 362.10 phr 80.75 vhr 1102.37 mmr 0 vhr_pmr 0.01 vhr

Compare Gaussian and skewed t-distribution fits

Gaussian fits

Gaussian QQ plots

Gaussian vs skewed t

Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
P_norm(X_min) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
P_norm(X_max) 0.546 0.580 1.599 0.796 1.161 0.746 0.793 0.903
P_t(X_min) 0.556 0.523 0.342 0.387 0.476 0.443 0.433 0.499
P_t(X_max) 0.448 0.469 1.135 0.614 0.739 0.518 0.543 0.613

Average number of years between min or max events (respectively):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
norm: avg yrs btw min 1.157446e+77 3.872665e+82 3.449915e+57 4.076881e+55 2.913573e+67 1.048138e+67 2.446247e+63 4.894371e+71
norm: avg yrs btw max 1.830960e+02 1.724260e+02 6.252800e+01 1.256870e+02 8.614100e+01 1.341240e+02 1.261390e+02 1.107200e+02
t: avg yrs btw min 1.798360e+02 1.912300e+02 2.924120e+02 2.584920e+02 2.099400e+02 2.257430e+02 2.309140e+02 2.002190e+02
t: avg yrs btw max 2.233340e+02 2.130540e+02 8.811500e+01 1.628680e+02 1.352750e+02 1.928930e+02 1.843230e+02 1.630560e+02

Lilliefors test

p-values for Lilliefors test.
Testing \(H_0\), that log-returns are Gaussian.

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
p value 0 0 0 0 0 0.001 0.001 0

Wittgenstein’s Ruler

For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?

Conditional probabilities for smallest observed log-returns:

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \leq x_{\text{min}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0.000 0.00 0.000 0.00 0.000 0.001 0.001 0.000
Prior prob 1.000 1.00 1.000 1.00 1.000 0.999 0.999 1.000
P[Gauss | Event] 0.838 0.82 0.768 0.69 0.815 0.486 0.418 0.965

Use \(1 - \text{p-value}\) from Lilliefors test as prior probability that the distribution is Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] = P_{\text{Gauss}}[X \geq x_{\text{max}}]\):

vmr vhr pmr phr mmr mhr vmr_phr vhr_pmr
Lillie p-val 0 0 0 0 0 0.001 0.001 0
Prior prob 1 1 1 1 1 0.999 0.999 1
P[Gauss | Event] 1 1 1 1 1 1.000 1.000 1

Velliv medium risk (vmr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.4145605 0.3807834

Objective function plots

Velliv high risk (vhr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.7391222 0.4858909

Objective function plots

PFA medium risk (pmr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.3304634 0.2764028

Objective function plots

PFA high risk (phr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

phr has the sstd fit with the highest sstd fit with thevalue of nu. Compare with other distributions:

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 2.0162301 0.4463226

Objective function plots

Mix medium risk (mmr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

mmr has the sstd fit with the lowest value of nu. Compare with other distributions:

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.3516393 0.2503782

Objective function plots

Mix high risk (mhr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.8775189 0.3400818

Objective function plots

Mix vmr+phr (vm_ph), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.7507161 0.3263777

Objective function plots

Mix vhr+pmr (mh_pm), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.540135 0.320090

Objective function plots

Velliv medium risk (vmr), June 2012 - April 2024

QQ Plot

Skewed \(t\)-distribution (sstd):

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

Monte Carlo

Sorted portfolio index values for last period of all runs

Convergence

Max vs sum

Max vs sum plots for the first four moments:

MC

IS

Skewed \(t\)-distribution with a normal proposal distribution.

Parameters

## [1] 1.4145605 0.3807834

Objective function plots